Completeness of metric spaces and existence of best proximity points

: In this paper, we discuss the existence of best proximity points of new generalized proximal contractions of metric spaces. Moreover, we obtain a completeness characterization of underlying metric space via the best proximity points. Some new best proximity point theorems have been derived as consequences of main results in (partially ordered) metric spaces.


Introduction and preliminaries
Fixed point theory has provided various tools to solve nonlinear functional equations in mathematics and many other related disciplines. In the context of metric spaces, one of the earlier fixed point theorems is the famous Banach contraction principle (shortly as BCP) [4] which has been applied to solve nonlinear operator equations (see [3] and references therein). BCP states that "if (Ω, ρ) is a complete metric space (shortly as C-MS) and a mapping F : Ω → Ω satisfies ρ(F u, F v) ≤ kρ(u, v) for all u, v ∈ Ω, and for some k ∈ [0, 1), then there is a unique point u in Ω such that u = F u, that is, F has a unique fixed point". There may arise situations where fixed points of mappings do not exist, for example, if A, B are non-empty subsets of Ω and F : A → B a nonself mapping, then u = F u may not have any solution. In such a situation, it is very useful to have a point u in A satisfying and if a point u in A exists that satisfies (1.1) is termed as a "best proximity point (BPP)" of F . Fan [15] discussed best approximation theorems in the context of normed spaces. For further generalizations of Fan's results, we direct the interested reader to [22,25]. Basha [6] generalized BCP for nonself mappings by proving BPP results for a new proximal contractions. Basha and Shahzad [8] extended these contractions and introduced proximal contractions of two different types and obtained best proximity points (BPPs). Samet et al. [24] initiated α − ψ−contractive type self-mappings which were further extended by Jleli and Samet [18] to the nonself contractive mappings along with the provision of results concerning the existence of singleton set of BPPs. Hussain et al. [16] initiated "modified (α, ψ)−proximal rational contractions" and some other useful results in this direction appeared in [20]. For more on the problem of existence of BPPs in various directions, we refer the readers to [1,7,[12][13][14]21].
We introduce a new set of proximal contractions of metric spaces and prove the existence of BPPs. Moreover, we also obtain metric completeness characterization via BPPs. As consequences of main results, we derive some important results in the literature as corollaries. We provide examples and show that some previous results are not applicable. Further, as applications, we obtain results corresponding to the main findings in the setup of "metric spaces equipped with a partial order".
Let (Ω, ρ) be a metric space and A and B non-empty subsets in Ω. Throughout this article, we use the following notations. 1) R + , R, N, N 0 for the set of nonnegative reals, reals, positive integers and nonnegative integers, respectively, 2) C(Ω) for the class of non-empty and closed subsets of (Ω, ρ), 3) ρ * (z, w) for ρ(z, w) − ρ(A, B) where z ∈ A and w ∈ B, 4) BPP(F ) for the set of BPPs of the mapping F : A → B and 5) F (F ) for the set of fixed points of the mapping F : A → B.
We introduce α p -proximal admissible mappings via a function p of A × A.
is a "Bianchini-Grandolfi gauge function (also known as (c)comparison function)" if ψ is non-decreasing, and there is l 0 ∈ N, and s ∈ (0, 1) implies We denote the set of such functions by Θ. The next lemma provides some useful characterizations of such functions.
The next BPP result is due to Abkar and Gabeleh [2] for Suzuki-type contractions.
for all w, z ∈ A. Further, if (A, B) has P P , then BPP(F ) is singleton.
Then BPP(F ) is non-empty.
Hussain et al. [17] presented the following result.
for all w, z ∈ A, and for some ψ ∈ Θ.
Then BPP(F ) is singleton.

Main results
The following is the first main result.
for some u 0 and u 1 in A 0 and a 4 F is continuous.
Proof. From the assumption (a 3 ), we have for some u 0 and u 1 in Continuing the process, a sequence {u n } in A 0 is obtained that satisfies That is Hence from (2.2) and (2.3), we get for all m > n > h. Thus {u n } is a Cauchy sequence in A. As A is a closed subset of (Ω, ρ) which is complete, so we get a u * ∈ A with u n → u * as n → ∞. By (a 4 ), F u n → F u * as n → ∞. This gives a contradiction. This proves the uniqueness.
We prove the next result without the assumption of continuity on F .
Then BPP(F ) is singleton.
Proof. On the similar lines as in Theorem 2.1, a sequence {u n } is obtained in A 0 that converges to u * ∈ A, and satisfies By (a 4 ), α(u n , u * ) ≥ 2 for all n ∈ N. Note that Hence (2.8) and (2.9) imply that hold for some n ∈ N, then we obtain and hold for some n ∈ N. Further, by (2.10) we get hold. If (2.11) holds for infinite many n ∈ N 0 , then using (1.2), we obtain as n tends to ∞, so as n tends to ∞, so we obtain a contradiction as ψ ∈ Θ. If That yields 2ρ(u n+1 , F u * ) ≤ ρ(u n , F u * ) + ρ(A, B).
On considering limit as n → ∞,we get implies u * is a BPP of F and the proof is complete. Consequently Similarly if (2.12) holds for infinite many n ∈ N 0 , then via (1.2), it follows that (2.14) Hence either (2.13) or (2.14) holds for all n ∈ N 0 . If we consider limit as n → +∞ in (2.13) and (2.14), we have either F u n → F u * or F u n+1 → F u * as n → ∞, that is, there is a subsequence {u n k } of {u n } with F u n k → F u * as k → ∞. Since u n k → u * as k → ∞, ρ(A, B) = lim k→∞ ρ(u n k +1 , F u n k ) = ρ(u * , F u * ).
The uniqueness of the BPP of F is followed on the similar lines as in Theorem 2.1.
Following example shows that Theorems 2.1 and 2.2 generalize some results properly in the literature. 2) It can be seen that Theorem 1.5 is not applicable here as for all u, v ∈ A, and for some r ∈ [0, 1) and ψ ∈ Θ.
Then BPP(F ) is singleton.
If ψ(x) = tx in Corollary 2.6, for 0 ≤ t < 1, we get the next result.
for all u, v ∈ A and for some t ∈ [0, 1).

Then BPP(F ) is singleton.
Above corollary yields another important result.
From Corollary 2.8, we fetch a result given in [2].
for all u, v ∈ A and for some t ∈ [0, 1), where Then BPP(F ) is singleton.
Corollary 2.10. Let (Ω, ρ) be a C-MS, A, B ∈ C(Ω) and a mapping F : A → B satisfying for all u, v ∈ A and for some t ∈ [0, 1). Further A 0 is non-empty, F (A 0 ) ⊆ B 0 and (A, B) has P P . Then BPP(F ) is singleton.

Completeness of metric spaces via the best proximity points
Completeness is an important property of metric spaces which is related to "end problem" in behavioral science (compare [5]). In a complete metric spaces, every Banach contraction has a fixed point but converse does not hold true. That means, there are incomplete metric spaces where every Banach contraction has a fixed point (see [11]). For more on completeness, we refer to [10]. In this section, we obtain completeness characterization via the existence of best proximity points.
If we set M(u, v) = ρ(u, v) in Corollary 2.8, we obtain the following corollary.
If we set A = B = Ω in Corollary 3.1, we get the following result.
Then F (F ) is singleton.
In the following theorem, we obtain completeness of metric space via the best proximity point theorem. For k ∈ (0, 1) and η ∈ 0, θ(k) 2 , let A k,η be a class of mappings F : A → B that satisfies (a) and (b) given below.
Let A * k,η be a class of mappings F : Ω → Ω that satisfies Let B k,η be a class of mappings F that satisfies (d) and (e) F (Ω) is denumerable, Then the statements 1-4 are equivalent: 1) The metric space (Ω, ρ) is complete.
and F x ∈ {u n : n ∈ N} (3.5) for all x ∈ Ω. From (3.5), we have g(F x) < g(x), hence F x x for all x ∈ Ω. That is, F (F ) is empty.

Best proximity point results in partially ordered metric spaces
Order structure is very important in connection with domain of words problem in computer sciences, equilibrium problems in economics (compare [23,28] and references therein) and many other related disciplines. In this section, with the help of the function α (used in the last section), and the main results in the last section, we deduce some important consequences related to the BPPs of nonself mappings of ordered metric spaces. Denote (Ω, ρ, ) by "partially ordered metric spaces" where ρ is a metric on Ω and a partial order on Ω.
If A = B, then F becomes nondecreasing mapping.
For a particular choice of the function ψ in Theorem 4.2, we get an important corollary as given below. for all u, v ∈ A with u v, and for some r ∈ (0, 1). Further A 0 is non-empty, F (A 0 ) ⊆ B 0 , (A, B) has P P and F is POP. Moreover u 0 u 1 and ρ(u 1 , F u 0 ) = ρ(A, B) for some u 0 and u 1 in A 0 . Then BPP(F ) is singleton. for all u, v ∈ A with u v, and ψ ∈ Θ. Further A 0 is non-empty, F (A 0 ) ⊆ B 0 , (A, B) has P P and F is POP. Moreover u 0 u 1 and ρ(u 1 , F u 0 ) = ρ(A, B) for some u 0 and u 1 in A 0 and for any non-decreasing sequence {u n } in A such that u n → u ∈ A as n → ∞, implies u n u for all n ∈ N. Then BPP(F ) is singleton.

Conclusions
This article dealt with the existence of best proximity points of generalized proximal contractions of complete metric spaces. We provided some examples to explain the main results and to show that the obtained results are proper generalizations of some existing results in the literature. Moreover, in this paper, a completeness characterization has been linked with the existence of best proximity points of mappings of metric spaces. One can consider the results in this paper for further study in the setup of more general spaces like b-metric spaces and quasi metric spaces. In quasi metric spaces, the problem of Smyth completeness via the existence of best proximity points of certain mapping would be worth doing.